ROLLING STONES WITH NONCONVEX SIDES I

J. Korean Math. Soc. 49 (2012), No. 2, pp. 265–291 http://dx.doi.org/10.4134/JKMS.2012.49.2.265

ROLLING STONES WITH NONCONVEX SIDES I: REGULARITY THEORY Ki-ahm Lee and Eunjai Rhee Abstract. In this paper, we consider the regularity theory and the existence of smooth solution of a degenerate fully nonlinear equation describing the evolution of the rolling stones with nonconvex sides: { M (h) = ht − F (t, z, z α hzz ) in { 0 < z ≤ 1 } × [0, T ] ht (z, t) = H(hz (z, t), h) on { z = 0 }. We establish the Schauder theory for C 2,α -regularity of h.

1. Introduction In this work, we are going to consider the wearing process of a rolling stone on a plane. Since the collision of a rolling stone on the plane causes the erosion of the surface, the speed of the erosion is proportional to the number of outward normal directions on a given surface area element, namely the Gauss curvature of the convex surface. Let us denote Σ be the surface evolving by the Gauss curvature flow and Σ∗ be the convex envelope of Σ which is the smallest convex surface containing Σ. Then any point P on the rolling stone Σ will propagate with the speed of Σ, ∂P ∗ (GCF) = K+ N, ∂t ∗ in the inward normal direction N , where K+ is the Gauss curvature of Σ∗ for ∗ P ∈ Σ ∩ Σ and otherwise zero. We denote by gij the metric and a second fundamental form of Σ and by gij and hij . We also denote the inverse of gij and hij by g ij and (h−1 )ij . The Weingarten map is given by hji = g jk hki and the eigenvalues, λ1 , . . . , λn , of hji are called principle curvatures. Then the Gauss curvature flow was introduced by Firey [14] and he showed that the smooth compact, strictly convex hypersurfaces with some symmetry shrinks to Received October 23, 2010. 2010 Mathematics Subject Classification. 35K20. Key words and phrases. rolling stone, degenerate fully nonlinear equations, free boundary problem. c ⃝2012 The Korean Mathematical Society

265

266

KI-AHM LEE AND EUNJAI RHEE

a round point. We will consider the case where the initial radial symmetric surface has a non-convex side and as a result the parabolic equation becomes degenerate along the interface of the non-convex surface and the convex surface. In this paper, we discuss the existence and regularity of the solution, and regularity of the free boundary. Let us assume that initially we have surface Σ = Σ0 ∪ Σ1 , where Σ0 is the non-convex side and Σ1 is the strictly convex part of the surface, Σ. The junction between the two sides is the (n − 1)-dimensional surface Γ = Σ0 ∩ Σ1 . Now we assume Σ0 is a concave graph z = φ(x) over a hyper plane. Since the equation is invariant under the rotation, we may assume the hyper plane is z = 0 plane and that Σ1 lies above this plane. The the lower part of Σ can be written as the graph of a function z = f (x) over a compact domain Ω ⊂ Rn on which the non-convex part can be written as a graph z = φ(x). Suppose z = f ∗ (x) be the convex envelope of the non-convex surface z = f (x). We can choose the domain Ω to be the set Ω = {x ∈ Rn : |∇f ∗ |(x) < ∞} so that f ∗ turns vertical at the boundary Γ . Let us denote by Tc the time when the area of the non-convex side Σo of the surface shrinks to zero. Since we only consider the surface symmetric with respect to z-axis, we may denote the lower part of Σ1 by z = f (r, t) for |x| = r and the non-convex part Σ0 by z = φ(r) for |x| = r. Note that f (r, 0) = φ(r) on Σ0 . Suppose z = f ∗ (r, t) be the convex envelope of the non-convex surface z = f (r, t). Then under the Gauss curvature flow, the envelope evolves as (1.1)

ft =

det(D2 f ∗ ) (1 + |Df ∗ |2 )

n+1 2

.

Let Ω(f ) = {(x, t) : |x| = r, f (r, t) = f ∗ (r, t)} and Ωt (f ) = {x : (x, t) ∈ Ω(f )}. The free boundary is denoted by Γ (f ) = ∂Ω(f ) and Γt (f ) = {x : (x, t) ∈ Γ (f )}. In particular, we denote Ωt = Ωt (f ) and Γt = Γt (f ). To understand the local behavior, we consider a simple model near the free boundary r = γ(t). (1.1) will be ( ∗ )n−1 ∗ frr fr ft = n+1 , I r where I = (1 + (fr∗ )2 ) 2 . Now we want to investigate whether the speed of propagation of the free boundary is non-degenerate and finite as it does in 1

ROLLING STONES WITH NONCONVEX SIDES

267

the flat spot case, [11]. fr will be zero on the free boundary otherwise it will propagate with infinite speed. Notice that on the free boundary Γ (f ), we have f ∗ (γ(t), t) = φ(γ(t), t). Then ft∗ + fr∗ γ ′ (t) = φr γ ′ (t), ft∗ γ ′ (t) = , φr − fr∗ and ft∗ = ft on Ω(t). These imply γ ′ (t) =

(1.2)

∗ (fr∗ )n−1 frr . rn−1 (φr − fr∗ )I n+1

fr = 0 otherwise it will propagate with infinite speed. If limr→γ(t)+ fr (x, t) > 0, ∗ then we have frr (γ(t), t) = ∞ which implies the speed of the propagation of the free boundary is ∞. If we expect the speed of the free boundary to be finite, fr∗ (γ(t), t) = limr→γ(t)+ fr (x, t) = 0. To find the behavior of fr away from the free boundary, let us try fr ≈ (r − γ(t))α0 for some αo . From the fact that γ ′ (t), φr (γ(t)), and I are of order one, it is easy to see αo = n1 and that we expect the optimal regularity of f to 1 be C 1, n . However, if we let the pressure g(r, t) = n1 frn ≈ (r − γ(t)), 1

g α grr g α gr g n gr2 gt = n−1 n+1 − (n − 1) n n+1 − (n + 1) n−1 n+3 , r I r I r I 2

1

n 2 where α = n−1 n , I = (1 + (ng) ) and we may expect better regularity for g as for the pressure of the porous medium equation [5]. Let us return back to the original equation with free boundary condition ( )n−1 f ∗ fr∗ (1.3) γ ′ (t) = rr (r, t) ∈ Γ (f ) φr r

since I = (1 + fr2 )(n+1)/2 = 1 on Γ (f ). Since g(γ(t), t) = 0, gt γ ′ (t) = − . gr With (1.2), we conclude gt = − and (GCFP)

gr2 φr rn−1

 { } 1  g n gr2 g α grr g α gr 1 gt = n+1 − (n − 1) − (n + 1) in Ω(g) I r n−1 rn r n−1 I n+3 2  gt (r, t) = − gn−1 r on Γ (g) φr r

and regularity of the free boundary Γt = ∂{x : g(r, t) = 0 } with finite speed 2 n+1 n 2 . of propagation where α = n−1 n and I = (1 + (ng) )

268


Inspired by [5], [10], [11], the proof of the existence of smooth solution of g is based on the idea of global change of coordinates by setting g(h(z, t), t) = z where {x : x = h(z, t)} is the level set of g. This transformation enables us to change the free boundary problem into an fixed boundary value problem { M (h) = ht − F (t, z, z α hzz ) in { 0 < z ≤ 1 } × [0, T ] (1.4) ht (z, t) = H(hz (z, t), h) on { z = 0 }. This equation is governed by metric s where d2 s =

dx2 2xα

and ∆s h = xα hxx which is no longer degenerating with respect to this new metric s. In this paper, we are going to prove the regularities of h at (1.4) and the existence of smooth solution. We establish Schauder estimate with respect to s for the model equation in Sections 2 and 3. To prove the existence of solution, we apply the inverse function theorem between certain Banach spaces in Section 4. The key lemma is the Schauder estimates for the degenerate equation which is a perturbation theory from the model equation when the coefficient are Hölder continuous. First let us summarize the notations. Notations: (1) The convex surface Σ = Σ0 ∪ Σ1 where Σ0 is the non-convex side and Σ1 is the strictly convex part of the surface, Σ. (2) f ∗ is the convex envelope of f which the supremum of all linear functions below f . (3) The domains will be defined as the followings: Ω(t) = {x ∈ Rn : |Df ∗ (x, t)| < ∞}, Ω(g) = {(x, t) : x ∈ Ω(t), 0 ≤ t < T, g(x, t) > 0}, Ωt = {x ∈ Ω(t) : g(x, t) > 0}, Ωtk = {x ∈ Ω(t) : 0 < g(x, t) ≤ k}, Ωt × [0, T ] = ∪0≤t≤T (Ωt × {t}) = Ω(g), Ωtk × [0, T ] = ∪0≤t≤T (Ωtk × {t}), Γ (g) = ∂{(x, t) : g(x, t) = 0}, Γt = {x : (x, t) ∈ Γ (g)} = ∂{x : g(x, t) = 0} and Γt is the graph of r = γ(t), S0 = {x > 0} and S = S0 × [0, ∞). Notice that Ω(t) = {x : g(x, t) = 0} ∪ Ωt and Ω(0) = Ω. 2−α Q+ ≤ t ≤ 1}. R = {(x, t) : 0 ≤ x ≤ R, 1 − R (4) The parabolic distance between two points P = (x1 , t1 ) and Q = (x2 , t2 ) is √ |x1 − x2 | s[P, Q] = α + |t1 − t2 |. α 2 2 |x1 + x2 |


269

(5) Dx f = fx and Dx f = xα fx . Dx2k f = (Dx Dx )k f and Dx2k+1 f = Dx (Dx Dx )k f . (6) The Hölder norms of f in a set A: )−f (Q)| ∥f ∥Cs0 (A) = supx∈A |f (x)|, ∥f ∥Hsγ (A) = supP ̸=Q∈A |f (P s(P,Q)γ , α ∥f ∥Hs2+γ (A) = ∥x fxx ∥Hsγ (A) + ∥ft ∥Hsγ (A) , ∥f ∥Cs2 (A) = ∥f ∥Cs0 (A) + ∥fx ∥Cs0 (A) + ∥xα fxx ∥Cs0 (A) + +∥ft ∥Cs0 (A) , ∑ i ¯ x,x Dtj f ∥Cs0 (A) , ∥f ∥Cs2k (A) = i+j≤k ∥D ∑ i j γ ∥f ∥Hsk+γ (A) = 2i+j=k ∥Dx Dt f ∥Hs (A) , ∥f ∥Cs2k+γ (A) = ∥f ∥Cs2k (A) + ∥f ∥Hs2k+γ (A) . (7) T0 g(x, t) = g(0, 1), T1 f (x, t) = f (0, 1) + fx (0, 1)x, T2−α,1 f (x, t) = f (0, 1) + fx (0, 1)x +

α 2−α 1 (2−α)(1−α) x fxx (0, 1)x

+ ft (0, 1)(t − 1),

Ri f = f − Ti f for i = 0, 1 and R2−α,1 f = f − T2−α,1 f . Let us start with showing the existence of the solution for short time by introducing a simple model equation in the following section. 2. Linear degenerate model equations In this section we are going to prove the regularity of the solutions of { L1 f = ft − xα fxx = g for x > 0 (2.1) B 1 f = ft − fx = 0 on x = 0. 2.1. Linearized equations To find the model equation above, let us introduce a new variable z representing the level of g and then the value r will be a function h(z, t) of (z, t) satisfying g(h(z, t), t) = z. Then, by taking the derivatives with respect to z and t respectively, we have 1 ht hzz gr = gt = − . grr = − 3 hz hz hz The equation (1.4) will transfer into [ { α }] 1 z hzz (n − 1)z α L(h) = ht − + = 0 for z > 0, J h2z hn−1 hn (2.2) 1 B(h) = ht − = 0 on z = 0, n−1 φr h hz where J = (1 + n1/n z 1/n )3/2 . Then the linearization L˜ of L around h is [ 1 1 ˜ 1 2z α hzz ˜ ˜ ˜ ˜ L(h) = ht − z α h − hz zz J h h2z J hh3z { α } ] 1 z hzz (n − 1)n z α ˜ − + h for z > 0, J h2 h2z hn+1

270


˜ =h ˜t + ˜ h) B(

˜z ˜z h (n − 1)h + =0 φr hn−1 h2z φr hn hz

on z = 0.

Let us consider the following equation our model equation L1 f = ft − xα fxx − g with a boundary condition B1 f = ft − fx − h. Since the diffusion is governed by metric s, the parabolic distance between two points P = (x1 , t1 ) and Q = (x2 , t2 ) is √ |x1 − x2 | s[P, Q] = α + |t1 − t2 |. α |x12 + x22 | Denote the box Bη = {0 ≤ x ≤ η 2−α , 0 ≤ t ≤ η }. We will denote by Csγ the space of Hölder continuous functions with respect to this metric s and Cs2+γ the space of function f such that xα fxx , ft and f in Csγ with norm ∥f ∥Cs2+γ (Bη ) = ∥f ∥Cs2 (Bη ) + ∥ft ∥Csγ (Bη ) + ∥xα fxx ∥Csγ (Bη ) . 2.2. Existence of smooth solution Theorem 2.1 (Existence and Uniqueness of Smooth Solutions). Assume that g is a smooth function with compact support on S = S0 × [0, ∞), which vanishes at t = 0. Then, there exists a unique smooth solution f of the initial value problem { L1 f = ft − xα fxx = g for x > 0 B1 f = ft − fx = h on x = 0. Moreover, for any T > 0 there exists a constant C(T ) depending only on T so that ||f ||C 0 (ST ) ≤ C(T ) (||g||C 0 (S) + ||h||C 0 (S0 ) ). Proof. Let’s first apply the Fourier-Laplace transform to convert the equation L1 f = ft − xα fxx = g to an ordinary differential equation with regular singular point at x = 0. Then xα fexx − (xα ξ 2 + τ )fe + ge = 0, where fe(x, τ ) =

∫

∞

e−tτ f (x, t)dt

t=0

and

∫ ge(x, τ ) =

∞

e−tτ g(x, t)dt.

t=0

Next we convert the boundary condition B(·, 0) = ft − fx = h and obtain τ fe(0, τ ) − fex (0, τ ) − e h(0, τ ) = 0,


where e h(0, τ ) =

∫

∞

271

e−tτ h(0, t)dt.

t=0

To show that the inverse Laplace transform is well defined, let us write fe = p + iq, ge = g1 + ig2 , e h = h1 + ih2 and τ = ρ + iσ, the ordinary differential equation satisfied by fe becomes equivalent to the system { xα pxx − ρp + σq = g1 xα qxx − ρq − σp = g2 with

{

px = ρp − σq − h1 qx = σp + ρq − h2

at x = 0. Then F = (p2 + q 2 )/2 satisfies the differential inequality xα Fxx − 2ρF ≥ pg1 + qg2 with Fx = 2ρF − (h1 p + h2 q) at x = 0. By Young’s inequality, ) ( 2 C C p + q2 − (g12 + g22 ) ≥ −ρ F − A(τ ) pg1 + qg2 ≥ −ρ 2 ρ ρ and

( −(ph1 + qh2 ) ≥ −ρ

p2 + q 2 2

) −

C 2 C (h1 + h22 ) ≥ −ρ F − A(τ ) ρ ρ

with A(τ ) = supx>0 (g12 + g22 )(x, τ ) + (h21 + h22 )(τ ). Hence, since ρ = xα ξ 2 + Re(τ ) ≥ Re(τ ) the function Fe = F −

C A (Re(τ ))2

satisfies the differential inequality xα Fexx − Re(τ ) Fe ≥ 0 and Fex ≥ Re(τ ) Fe. Moreover Fe is smooth and bounded on x > 0, for all real ξ and complex τ with Re(τ ) > 0. It follows from the maximum principle that Fe ≤ 0, for all x > 0 which gives us the bound |fe(x, τ )| ≤

C (sup |e g |(x, τ ) + |e h|(τ )) Re(τ ) x>0

272


with C an absolute, positive constant. Since supx>0 |e g |(x, τ ) and |e h|(τ ) decay rapidly as |τ | → ∞ with Re(τ ) > 0, it follows from this estimate that the function f given by ∫ +i∞+ϵ f (x, t) = lim etτ fe(x, τ )dτ ϵ→0

−i∞+ϵ

is well defined and therefore a smooth solution of the equation L1 f = g with B1 f = h at x = 0. For large enough τ , Fe ≤ 0 and F ≤ C B2 . As |ξ| → ∞, |τ | → ∞, fe decays rapidly as e h decays. It is also easy to see that ||f ||C 0 (ST ) < ∞ for all T > 0. In addition, for any positive integer n, we have ∫ +i∞+ϵ n etτ τ n fe(x, τ )dτ. Dt f (x, t) = lim ϵ→0

−i∞+ϵ

Therefore, denoting by L the Laplace transform, we have L(Dtn f )(x, τ ) = τ n L(f )(x, τ ) for all τ > 0. This immediately implies that Dtn f (x, 0) = 0 for all positive integers n, making f a smooth function on S with f (·, 0) = 0. This answers the existence question. □ 2.3. Local derivative estimates In this section, we will prove certain local estimates on the derivatives of f . Lemma 2.2. If f is a smooth solution of (2.1) and if |f |+R2−α |g|+R1−α |gx | < + CB 2−α B in Q+ ≤ t ≤ 1}. 2R , then |fx | < R in QR = {(x, t) : 0 ≤ x ≤ R, 1 − R Proof. Let us scale the function f¯(x, t) = f (Rx, R2−α t) and then f¯ satisfies { L1 f¯ = f¯t − xα f¯xx = R2−α g for x > 0 (2.3) f¯t − R1−α f¯x = 0 on x = 0. For the simplicity, we will replace f¯ by f from now on. Let X = B(1+f 2 )+ηfx and let us assume that the maximum of X on [0, T ] has been achieved at (xo , t0 ). When (xo , to ) is an interior point, we have Xx = 2Bf fx + η 2 fxx + ηx fx = 0 at (xo , to ).


273

X will have the following contradiction.

(2.4)

0 ≤ Xt − xα Xxx 1 = − fx [2Bx1+α ηfx − xηηt xη ( ) + xα (−4B(1 + f )x + αη)ηx − 2xηx2 + η (2B(1 + f )α + xηx,x ) ]

+ 2B(1 + f )g + ηgx ) C ( 1+α 2 Bx ≤ − ηfx + αxα Bη − x1+α Bηx < 0 xη for large B > 0 by choosing η such that ηx < 0. When xo = 0, 0 ≥ xα Xx = Rηfx > 0. Similarly we can find the lower bound of X.

□

Lemma 2.3. If f is a smooth solution of (2.1) and if |f | + R|fx | + R2−α |g| + + CB α R1−α |gx | + |xα gxx | < B in Q+ 2R , then |x fxx | < R2−α in QR = {(x, t) : 0 ≤ 2−α x ≤ R, 1 − R ≤ t ≤ 1}. Proof. As the lemma above, f¯(x, t) = f (Rx, R2−α t) satisfies (2.3). For the simplicity, we will replace f¯ by f for now. Let us consider the quantity X = A(B + xα/2 fx )2 + η 2 (xα fxx )2 and assume that the maximum of X on [0, T ] has been achieved at (xo , t0 ). 1−α For (xo , to ) ∈ ∂0 Q+ fx which means X is bounded by a uniform R , ft = R constant from Lemma 2.2. When (xo , to ) is an interior point, we have ( ) 1 fxxx = 2 x−1−2α [−Ax2β βfx2 − Axβ fx Bβ + x1+β fx,x η fx,x (2.5) ( ) − fx,x ABx1+β + x2α η (αη + xηx ) fx,x ] from Xx = 0. From a simple computation, we will have 1 0 ≤ Xt − xα Xxx ≤ 2−α 2 2 [−Ax2−α η 2 X 2 x η X (2.6)

+ AαηxX 3/2 − Axα/2 X − A2 xαX 1/2 − xα α2 A2 ] α 1 ≤ 2−α 2 2 [−A(ηx1− 2 X − C1 )2 x η X − Axα/2 (X 1/2 − C2 )2 − xα (α2 A2 − C3 xA3 )]

which will be negative for 0 < x < δ by selecting a large A and a small δ > 0 and which will be also negative for 0 < x < 2 for large X. Therefore we have a contradiction. □ ¯ x f = xα Dx , Now let us introduce higher derivatives. Set Dx f = fx , D α ¯ ¯ and Dxx f = Dx Dx f = x Dxx f . By applying Lemmas 2.2 and 2.3 on Dtk u inductively, we have following estimate.

274


Corollary 2.4. Let us assume that f is a smooth solution of (2.1) and that |f | < B on Q+ 2r . Then we have CB k ¯ xx |D f | ≤ k(2−α) r in Q+ r . 3. Polynomial approximation To obtain Schauder estimates for L1 f = ft − xα fxx = g (3.1) B1 f = ft − fx = 0

for x > 0, on x = 0,

we need to prove some polynomial approximation theories. Theorem 3.1 (Cycloidal Polynomial Approximation Theorem). There exists a constant C with the following property. For every smooth function f on the box Bs such that B1 f = 0 on x = 0, we can find a polynomial b x2−α of degree 2 − α in space and one in time p(x, t) = a + b t + b x + (2−α)(1−α) so that for every r ≤ s [( ) ] r 3−α 2−α ∥f − p∥r ≤ C ∥f ∥s + s ∥L1 f ∥s . s Proof. Now we set h to be a replacement of f satisfying a homogenous equation: L1 h = ht − xα hxx = 0 for x > 0, (3.2)

B1 h = ht − hx = 0

on x = 0, on {x > 0} ∩ ∂p Bs .

h=f

To find the different between f and h, let k = f − h and then k satisfies the equation (3.1) with zero on {x > 0} ∩ ∂p Bs . A comparison k with a super-solution k + = (s2−α − |x|2−α ), tells us ∥k∥s ≤ Cs2−α ∥g∥s . Let p be a 2nd order Taylor polynomial of k, ( ) ( ) ( ) ( ) ( ) x 0 0 0 2−α 0 p =h + hx x + (xα hxx ) x + ht (t − 1) t 1 1 1 1 at the point ( 01 ). Then ∥h − p∥r ≤ C

( r )3−α s

∥h∥r .

Since f = h + k, we have ∥f − p∥r ≤ ∥h − p∥r + ∥k∥r [( ) ] r 3−α ≤C (∥h∥r ) + s2−α ∥g∥s (3.3) s ] [( ) r 3−α ∥f ∥s + s2−α ∥L1 f ∥s . ≤C s

□


275

Theorem 3.2 (Schauder Estimate). For each β in 0 < β < 1 there exists a constant S with the following property. If f is a smooth function on the box B1 such that T2−α,1 f = 0 in x > 0 and B1 f = 0 on x = 0, then ( ) ∥f ∥r ∥L1 f ∥r sup ≤ C ∥f ∥ + sup . 1 2−α+β rβ 0 0, the initial value problem   in ST Lf − cf = g f (·, 0) = f 0 on S0   B1 f = 0 on x = 0 admits a unique solution f ∈ Csk,2+β (ST ) which satisfies the estimate ( ) ∥f ∥Csk,2+β (ST ) ≤ C(T ) ∥f 0 ∥Csk,2+β (S0 ) + ∥g∥Csk,β (ST ) for some constant C(T ) depending on k, β, c and T . Proof. For each f ∈ Csk,β (S) there is a solution f˜ ∈ C k,2+β (S) such that  ˜  in ST Lf = cf + g 0 ˜ f (·, 0) = f on S0   ˜ B1 f = 0 on x = 0. Let C k = {f ∈ Csk,β (S)|f˜(·, 0) = f 0 on S0 and B1 f˜ = 0 on x = 0} and then C k is convex. The Schauder estimate, Corollary 3.17, says T f = f˜ maps C k to a precompact subset of C k . Now the Schauder fixed point theorem says there is f ∈ C k such that T f = f , which is equivalent to the conclusion above. □ 4. Degenerate equations with variable coefficients In this section we extend the existence and uniqueness theorem to quasi linear degenerate equations and linear degenerate equations. First we consider the linear degenerate equations of the form { Lw = wt − (a xα wxx + b w) = g in x > 0 (4.1) Bf = ft − c(t)fx = ψ(t) on x = 0

286


on the cylinder Ω × [0, ∞), where Ω is a compact domain in R with smooth boundary. We assume the coefficient a strictly positive and all coefficients a and b belong to appropriate Hölder spaces which will be defined later. When the boundary is flat and the coefficients are constants, this equation takes the form of the model equation studied in Section 2 { ft − xα fxx = g in x > 0 (4.2) ft − fx = 0 on x = 0 on the half-space x ≥ 0. Imitating the model case where the operators are defined on the half space {x ≥ 0}, we define the distance function s in Ω. In the interior of Ω the cycloidal distance will be equivalent to the standard Euclidean distance, while around any point P ∈ Γ , s is defined as the pull back of the cycloidal distance on the half space S = {x ≥ 0}, as defined in Section 1, via a map Φ : S → Ω that straightens the boundary of Ω near P . The parabolic distance in the cycloidal metric is equivalent to the function [( ) ( )] √ x1 x s¯ , 2 = s(P1 , P2 ) + |t1 − t2 |. t1 t2 Now suppose that A is a subset of the cylinder Ω × [0, ∞) which is the closure of its interior. As in Section 1, we denote by Csγ (A) the space of Hölder continuous functions on A with respect to the metric s and by Cs2+γ (A) the space of all functions w on A such that w, wt , wi and xα wxx , extend continuously up to the boundary of A and the extensions are Hölder continuous on A of class Csγ (A). They are both Banach spaces under the norms ∥w∥Csγ (A) , and ∥w∥Cs2+γ (A) = ∥w∥C 2 (A) + ∥xα wxx ∥Csγ (A) + ∥wt ∥Csγ (A) . Also, we denote by Cs2k,γ (A) and Cs2k,2+γ (A) the spaces of all functions w i ¯ xx ¯ xx w = xα Dx2 w and i + j = 2k Dtj w with D whose 2k-th order derivatives D γ 2+γ exist and belong to the spaces Cs (A) and Cs (A) respectively. Both spaces equipped with the norms ∑ ∑ ¯ i Dtj w∥C γ ( A) + ¯ i w∥C 0 ( A) ∥w∥Cs2k,γ (A) = ∥D ∥∇x D xx x,x s i+j≤k

and ∥w∥Cs2k,2+γ (A) =

i 0, the set Ωσ = { x ∈ Ω : dist (x, Γ ) ≥ σ } by QT , for T > 0, the cylinder Ω × [0, T ], we can now state: Theorem 4.1 (Existence and Uniqueness). Let Ω be a compact domain in R with smooth boundary and let k be a nonnegative integer, a a number in 0 < a < 1 and T a positive number. Assume that the coefficients a, b and c of the operator L belong to the space Cs2k,β (QT ) and satisfy the ellipticity condition a ≥ λ > 0, c ≥ λ > 0 and

1 λ for some positive constants λ. Then, given any function w0 ∈ Cs2k,2+γ (Ω) and any function g ∈ Cs2k,γ (QT ) there exists a unique solution w ∈ Cs2k,2+γ (QT ) of the initial value problem   in QT Lw = g wt (r, 0) = wr (r, 0) on Ω   Bw = φ on ∂0 QT = ∂QT ∩ {x = 0} ∥a∥Cs2k,γ (QT ) + ∥b∥Cs2k,γ (QT ) + ∥c∥Cs2k,γ (∂0 QT ) ≤

satisfying ∥w∥

Cs2k,2+γ (QT )

( ≤ C(T ) ∥w0 ∥

Cs2k,2+γ (Ω)

+ ∥g∥

Cs2k,γ (QT )

+ ∥φ∥

) Cs2k,γ (∂0 QT )

.

The constant C(T ) depends only on the domain Ω and the numbers γ, k, λ, σ and T . Proof. For small enough δ > 0, we begin by expressing the compact domain Ω as the finite union Ω = Ω0 ∪ ( ∪ Ωl ) l≥1

of compact domains in such a way that dist (Ω0 , Γ ) ≥

ρ 2

> 0 and for l ≥ 1

Ωl = Bρ (xl ) ∩ Ω with Bρ (xl ) denoting the ball centered at xl ∈ Γ of radius ρ > 0. The number ρ > 0 will be determined later. Note that the operator L restricted on the interior domain Ω0 is L1 and non-degenerate. Therefore, the Schauder theory for linear parabolic equations implies that L is invertible when restricted on functions which vanish outside Ω0 . Here our Hölder spaces with respect to the cycloidal metric s on the interior domain Ω0 is the standard Hölder spaces, where the Schauder theory holds. Next, look into the domains Ωl , l ≥ 1, close to the boundary of Ω. Denoting by B the half unit ball B = {x ∈ B1 (0) ; x ≥ 0 }

288


and by Qδ the cylinder

Qδ = B × [0, δ] we select smooth charts Υl : B → Ωl , which flatten the boundary of Ω, i.e., they map B ∩{x = 0} onto Ωl ∩∂Ω and have Υl (0) = xl for ρ chosen sufficiently small. □ Under the change of coordinates induced by the charts Υl , the operators L and B, restricted on each Ωl × [0, δ], is transformed to operator L¯l and B of the form L¯l w ¯=w ¯t − (¯ al xα w ¯xx + ¯bl w) ¯ defined on B × [0, δ] and Bl w ¯=w ¯t − c¯w ¯x defined on (B ∩ {x = 0}) × [0, δ] respectively. Moreover, the charts Υl can be chosen appropriately so that the coefficients of L¯l are in Csk,γ . The continuity of the coefficients then implies that the constant coefficient operator Lel w ¯=w ¯t − [¯ al xα w ¯xx + ¯bl w ¯] having a ¯l = al (0), ¯bl = ¯bl (0) when defined on Qδ = B × [0, δ] has coefficients sufficiently close to the coefficients of L¯l in the space Csk,γ (Qδ ), if ρ and δ are chosen sufficiently small. Combining theorem for the model equation with the perturbation argument, we can give the generalization of the local Schauder estimates for variable coefficient equations. For simplicity we will assume that the operator L˜ has the form ˜ = wt − (a xα wxx + bw) Lw defined on the half space x ≥ 0 B˜l w = wt − cwx defined on x = 0. As at the beginning of Section 2, we define the box of side r around a point P = ( xt00 ) and let Br be the box around the point P = ( 01 ). We have the following theorem: Theorem 4.2. Assume that the coefficients a and b of the operator L belong to the space Csγ (B1 ), for some number γ in 0 < γ < 1 and satisfy a ≥ λ > 0,

c≥λ>0

and ∥a∥Csγ (QT ) + ∥b∥Csγ (QT ) + ∥c∥Csγ (∂0 QT ) ≤ 1/λ and for some positive constants λ. Then, there exists a constant C depending only on γ, λ such that ( ) ˜ ∥C γ ( B ) + ∥Bf ˜ ∥C γ (∂ B ) ∥f ∥Cs2+γ ( B1/2 ) ≤ C ∥f ∥Cs◦ ( B1 ) + ∥Lf 1 0 1 s s for all functions f ∈ Cs2+γ ( B1 ).


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Proof. As in [5], we will assume that f is a C ∞ function on B1 . The case f ∈ Cs2+γ (B1 ) will then follow via a standard approximation argument, using the smoothing operators. □ The next result follows from the Schauder estimate. Theorem 4.3. Under the same hypotheses as in Theorem 3.2 and for any number r ≤ 1 there exists a constant C(r) so that ) ( ˜ ∥C γ (B ) . ∥f ∥Cs2+γ ( Br/2 ) ≤ C(r) ∥f ∥Cs◦ ( Br ) + ∥Lf r s Now we consider the quasi-linear degenerate equations of the form wt = xα F (t, xw, Dw) wxx + G(t, x, w, Dw) on the cylinder QT = Ω × [0, T ], T > 0. Lets denote by P the operator P w = x F (t, x, w, Dw) wxx + G(t, x, w, Dw) and by M the operator M w = wt − P w. Then, if w ¯ is a fixed point in Cs2+γ (QT ), the linearization of the operator M f(w) at the point w ¯ is the operator M ˜ = DM (w)( ¯ w) ˜ =w ˜t − DP (w)( ¯ w) ˜ with DP (w)( ¯ w) ˜ = xα F (t, x, w, ¯ Dw) ¯ w ˜xx + [ xα Fwl (t, x, w, ¯ Dw) ¯ w ¯xx + Gwl (t, x, w, ¯ Dw) ¯ ]w ˜l + [xα F w (t, x, w, ¯ Dw) ¯ w ¯xx + Gw (t, x, w, ¯ Dw) ¯ ] w. ˜ Using the Inverse Function Theorem with the theorem above, this implies the following initial value problem is solvable: Theorem 4.4. Assume that Ω is a compact domain in R with smooth boundary and let k be a nonnegative integer, and 0 < γ < 1, T > 0 positive numbers. Also, let w0 be a function in Csk,2+γ (Ω). Assume that the linearization DM (w) ¯ of the quasi-linear operator M w = wt − xα F (t, x, w, Dw) wxx − G(t, x, w, Dw) defined on QT = Ω ×[0, T ], satisfies the hypotheses of Theorem 3.2 at all points w ¯ ∈ Csk,2+γ (QT ), such that ∥w ¯ − w0 ∥Csk,2+γ (QT ) ≤ µ, µ > 0. Then, there exists a number τ0 in 0 < τ0 ≤ T depending on the constants γ, k, λ and µ, for which the boundary value problem { wt = xα F (t, x, w, Dw) wxx + G(t, x, w, Dw) in Ω × [0, τ0 ] wt (·, 0) = H(t, x, wwx ) on x = 0 admits a solution w in the space Csk,2+γ (Ω × [0, τ0 ]). Moreover, ∥w∥Csk,2+γ (Ω×[0,τ0 ]) ≤ C ∥w0 ∥Csk,2+γ (Ω) for some positive constant C which depends only on γ, k, λ and σ. Acknowledgement. Ki-Ahm Lee was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD, Basic Research Promotion Fund) (KRF-2008-314-C00023).

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References [1] B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. [2] D. Chopp, L. C. Evans, and H. Ishii, Waiting time effects for Gauss Curvature Flow, Indiana Univ. Math. J. 48 (1999), no. 1, 311–334. [3] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. , On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. [4] Pure Appl. Math. 44 (1991), no. 4, 469–483. [5] P. Daskalopoulos and R. Hamilton, The free boundary in the Gauss curvature flow with flat sides, J. Reine Angew. Math. 510 (1999), 187–227. , The free boundary for the n-dimensional porous medium equation, Internat. [6] Math. Res. Notices 1997 (1997), no. 17, 817–831. [7] , Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998), no. 4, 899–965. [8] , C ∞ -regularity of the interface of the evolution pp-Laplacian equation, Math. Res. Lett. 5 (1998), no. 5, 685–701. [9] P. Daskalopoulos, R. Hamilton, and K. Lee, All time C ∞ -regularity of the interface in degenerate diffusion: a geometric approach, Duke Math. J. 108 (2001), no. 2, 295–327. [10] P. Daskalopoulos and K. Lee Free-Boundary Regularity on the Focusing Problem for the Gauss Curvature Flow with Flat sides, Math. Z. 237 (2001), no. 4, 847–874. [11] , Worn stones with flat sides all time regularity of the interface, Invent. Math. 156 (2004), no. 3, 445–493. [12] , H¨ older regularity of solutions of degenerate elliptic and parabolic equations, J. Funct. Anal. 201 (2003), no. 2, 341–379. [13] P. Daskalopoulos and E. Rhee Free-boundary regularity for generalized porous medium equations, Commun. Pure Appl. Anal. 2 (2003), no. 4, 481–494. [14] W. Firey, Shapes of worn stones, Mathematica 21 (1974), 1–11. [15] R. Hamilton, Worn stones with flat sides, A tribute to Ilya Bakelman (College Station, TX, 1993), 69–78, Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994. [16] H. Ishii and T. Mikami, A mathematical model of the wearing process of a nonconvex stone, SIAM J. Math. Anal. 33 (2001), no. 4, 860–876. [17] , A level set approach to the wearing process of a nonconvex stone, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 53–93. [18] N. V. Krylov and N. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175. [19] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [20] K.-A. Lee and E. Rhee, Rolling Stones with nonconvex sides II: All time regularity of Interface and surface, preprint. [21] K. Tso, Deforming a hypersurface by its gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. [22] L. Wang, On the regularity theory of fully nonlinear parabolic equations I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27–76. [23] , On the regularity theory of fully nonlinear parabolic equations II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141–178.


Ki-ahm Lee Department of Mathematics Seoul National University Seoul 151-747, Korea E-mail address: [email protected] Eunjai Rhee Department of Mathematics Seoul National University Seoul 151-747, Korea E-mail address: [email protected]

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